"In science there is only physics, rest are only stamp collecting"
"Mathematics is like a tree. It spreads with the branches and parts , which are dependent on other parts. And in this fashion it gets simple enough to wholly rest on a few principles. These are the roots of the tree"
Quantum mechanicsSchrodinger equation | Matrix mechanics | Dirac Equation | Quantum electrodynamics
Theory of relativitySpecial theory of relativity | General theory of relativity
Matrix mechanics pdf
This is most modern and precise theory of atomic phenomena. Matrix mechanics reveals that the occurances that happen inside atom are probalistic. We have to work with a collection of data instead of certain changes governed by cause and effect relationship. The position observable is a matrix which has an infinite number of elements each of which is a possible measureable quantity. Momentum observable is also a matrix. What we can best expect is that certain change is more likely than the other. Only statistical knowledge about the atom can help us determine which change is more likely than other. Thus a new kind of mathematics has been developed , which compels us to describe everything about the atom in terms of probability. This kind of uncertainty is inherent in the quantum reality. Heisenberg picture of atom is related to operators A and the Hamiltonian H which is itself an operator. The time evolution of such operator A is the basic equations of motion according to Heisenberg;
Eigenvector is a vector which produces the same vector multiplied by some constant when it is acted by some linear operator like a matrix A. In quantum mechanics the operator H (hamiltonian) acts on wave function eigenvector
and produces a corresponding eigenvalue E.
Heigenberg's theory of atom treats subatomic particles differently than ordinary quantum theory. According to this theory atom or electron has no immediate reality as the objects of sense but only those properties ascribed to light wave. The troubles came , from trying to picture the atom or electron as in ordinary space. These entities must not be said to be in definite position at some time. If we want to retain the corpuscular model of electron , we can do it not assigning definite position to an electron. it must be replaced with a group of physical entities which represent the place of the electron. These quantities can be called physically observable quantities or radiations in the form of light.
Heisenberg's Uncertainty relation
The atomic model of Bohr had limitation due to its hypothetical orbits which can not be observed. Heisenberg thought this as a nonphysical description of the atom. So Heigenberg made a reasoning that the radiation that the electron emits can be taken as a physical observable. According to Bohr's theory an electron jumps from one orbit to another when it emits or absorbs light or electromagnetic energy without being in the intermediate space. The frequency of emitted radiation from an atom can be represented by two variable m and n , which represent two Bohr's orbits respectively. Comparing it with classical harmonic motion , which can be represented by Fourier series, Heisenberg showed that two observables x(t), p(t) can be correlated with this Fourier series. Each of these observables is a matrix (row m and column n). Thus atom can be completely modeled by two matrices : position and momentum matrices (with m rows and n columns) with frequency of radiation determined by plank's formula.
The Fourier series representation of harmonic motion is :
The equation of motion reasoned by Heigenberg has the following Fourier components which itself must oscillate at frequency W(mm) :
Observable quantities like phase, frequency and amplitude of radiation thus enter into the mathematical equation. But as the those are the quantities that depends on difference of two number, the position and momentum will be matrices of m rows and n columns. Sum of all the matrix elements thus has no physical significance as it represents just a wave.
Momentum and position matrix P(0), X(0)At certain moment (t) the position and momentum has to take values as matrices:
As it is evident that two matrices does not always commute , there is no well defined position and momentum of any particle at the same instant. The order of product of two matrices changes
the value of multiplication. If two matrices each have the same identical elements then the order of their product do not change the value of multiplication.
In case with the multiplication of vector with a matrix the resultant value of the multiplication is also a vector:
If each contains distinct elements , the order of multiplication changes the value of multiplication. As position and momentum, in general , do not contain identical elements , their product depend s on their order of multiplication. Such was the reasoning of Heisenberg. No state possesses definite position and momentum simultaneously. The next is general result of such anti-commutation. The result of the difference is number i (imaginary) multiplied by plank constant.
The product of variance of position and momentum , σ(x) and σ(p) can be proven to be less than h/2. This is the uncertainty that saves quantum mechanics from breaking down. A proof using standard deviation of operators can be made simply:Any operator O that acts on wave wave function ψ must have standard deviation σ
More compact and popular version of the uncertainty principle is :
There is some misconception about the uncertainty principle. This is not related to limitation of the precision of our observing instruments. Now matter how precise our measuring intruments are , there will be always some uncertainty or unpredictability. The uncertainty principle is intrinsic property of quantum world. This is the way our world in the smallest scale behaves. Due to this uncertainty principle particle and anti -particle always appear and annihilate in vacuum out of nothing. Classical objects like billiard balls, moon have definite position and momentum. We can determine their position and momentum simultaneouly. But when we try to measure the position of tiny objects like electron the very act of observation changes the momentum of the particle. Some momentum of the incident light is transferred to the electron. This change is random. As a result, we can not predict with certainty the exact momentum of the electron at the same time we measure its position.
Free Schrodinger equationSchrodinger equation plays a vital role in quantum mechanics. We set potential energy equal to zero in free Shcrodinger equation. The solution of such equation will then change identically.
Heisenberg's equation of motionHeiseberg's equation of motion describes the time evolution of operator A in heisenberg's picture. In heisenberg's picture operators carry time dependence while the state vector remains constant. The proof of equation's of motion is as follows:
The time derivative of operator O(s) is zero which is omitted in the first line. Terms:
O(s) = operator is Schroinger picture
O(h) = Operator in Heisenberg's picture.
U = unitary operator desrcibing time evolution.
Some useful equations in quantum mechanicsErhenfest's theorem :
View this video for more explanation:
Notes and additional comments
expectation value of a random experiment is the average value of all the outcomes of the experiment. Suppose you throw a six headed dice one time. Then the expectation value will be (1+2+3+4+5+6)/6 = 3.5. Similarly operators in quantum mechanics have expectation value. Operators act on observables to give outcome of quantum measurements. So it is natural that they have expectation values."there are some corners of the universe which have breed the most terrible things. Things that work against everything we believe in. These things must be fought "
Weirdness of quantum world
Quantum world is bizzare and mysterious. As for the atom, it gives us no clues of its existence untill it gives off radiation into outer world. The radiations come into discrete chunks of energy. That was the theory of Max Planck. If Max Planck did not invent his law we would be burned by the sun's ultraviolate rays. The smallness of planck's constant h makes the quantum world behaves differently than classical world.
Vector, scalar and tensor harmonics on three sphere are introduced in order to study gravitational physics. These harmonics are the eigenfunctions of covariant laplace operator which satisfy some divergence and trace identity ,and ortho-normality conditions.
The wave function is computed using these harmonics and corresponding Schrodinger equation can be formulated easily.
De sitter metric is a solution of Einstein's field equations. It was found by Astronomer De sitter. It is given as a static form which is like Swardchild's solution in case of non-rotating massive objects.
like Swardchild solution de sitter metric contains a singularity at some r -value. The n-dimensional de Sitter space, denoted by dS(n), is the Lorenzian manifold which is the analog of n-sphere(with its Riemanian metric), and which has positive constant curvature and simply connected for at least n=3.
Quantum mechanics have been merged with special relativity successfully. This apparent marriage has brought out some difficulties. Qantum mechanics is regarded as a complete and most succesful theory. But this theory is not free of contradiction. It has resulted in a paradoxical phenomena called measurement paradox. Measuring a quantum system is depenent on the system measuring it. Observing a particular quantum phenomena does not imitate classical behavior. Prior to measurement the system posesses a myriad possibilities to be evident when measured. This is so called measurement paradox.
On the other hand when special relativity is combined with quantum mechanics , infinities creep into the calculution. It is not desirable to work with infinties in real world situation. In general theory of relativity singularities arise because of very dense mass and energy distribution. Laws of physics breaks down and can not predict anything in such a case.
wheeler Dr with equation is a functional differential equation to combine features of quantum mechanics with General relativity. A functional differential equation is a equation which relates a derivative of a functional to the change of the function on which the functional depends.
Einstein said that "God does not play dice" as he was an opponent of quantum mechanics. Stephen Hawking claimed that God not only play dice but He throws them where we can not see them. Here is a visualization of black hole
from where nothing can escape. Everything else which enters a black hole's event horizon becomes lost inside the singularity.